Douglas A.Ruby

Basic Definitions

Relative Prices

A Producer Optimum

Equilibrium Analysis

Overview of Economics

Macroeconomic Principles

Microeconomic Principles

Macroeconomic Theory

Microeconomic Theory
Production and Production Possibilities

The Production Function
Production refers to the conversion of inputs, the factors of production, into desired output. A production function for a particular good or service is often written as follows:

Xi = f(L,K,M,R)

where Xi is the quantity produced of a particular good or service and:

• L represents the quantity and ability of labor input available to the production process.
• K represents capital input, machinery, transportation equipment, and other types of intermediate goods.
• M represents land, natural resources and raw material inputs for production, and
• R represents entrepreneurship, organization and risk-taking.

A positive relationship exists among these inputs and the output such that greater availability of any of these factors will lead to a greater potential for producing output. In addition, all factors are assumed to be essential for production to take place. The functional relationship f(.) represents a certain level of technology and know how, that presently exists, for conversion these inputs into output such that any technological improvements can also lead to the production of greater levels of output.

Production in the Short Run
In order to better understand the technological nature of production, we distinguish between short run production relationships where only one factor input may vary (typically labor) in quantity holding the other factors of production constant (i.e., capital and/or materials) and the long run where all factors of production may vary. The short run allows for the development of a simple two variable model to understand the behavior between a single variable input and the corresponding level of output. Thus we can write:
Xi = f(L;K,M,R)
or
Xi = f(L)

For example we could develop a short run model for agricultural production where the output is measures as kilograms of grain and labor is the variable input. The fixed factors of production include the following:

• 1 plow
• 1 tractor.... capital
• 1 truck
• 1 acre of land
• 10 kilograms of seed grain
We might hypothesize the production relationship to be as follows:

Table 1
(Constant Marginal Productivity)
 Input(L) Output(Xgrain) MPL 0 0 kg - 1 100 100 2 200 100 3 300 100 : : 100 10 1000 100

In this example we find that each time we add one more unit of labor, output increases by 100 kg. The third column MPL defines this relationship. This column measures the marginal productivity of labor -- a measure of the contribution of each additional unit of labor input to the level of output. In this case, we have a situation of constant marginal productivity which is unrealistic with production in the short run. Constant marginal productivity implies that as labor input increases, output always increases without bound -- a situation difficult to imagine with limited capital and one acre of land.

A more realistic situation would be that of diminishing marginal productivity where increasing quantities of a single input lead to less and less additional output. This property is just an acknowledgment that it is impossible to produce an infinite level of output when some factors of production (machines or land) fixed in quantity. Numerically, we can model diminishing marginal productivity as follows:

Table 2
(Diminishing Marginal Productivity)
 Input(L) Output(Xgrain) MPL 0 0 kg - 1 100 100 2 180 80 3 240 60 4 280 40 5 300 20 6 300 0

In this case, additional labor input results in additional output. However, the contribution of each additional unit of labor is less than previous units such that the sixth unit of labor contributes nothing to output. With 5 or 6 workers, the available amount of land cannot support additional output.

A short run production relationship can be modeled in the diagram below. In this example, labor is the variable factor input and land, capital, and entrepreneurship are fixed in quantity. There is a positive relationship between labor input and output levels, however, as additional labor in used, less and less additional output is produced (click on the second button). The shape of this production function is consistent with the law of diminishing marginal productivity.

Figure 1
Original Position

An Increase
in Labor Input

An Increase
in Capital Input

Changes in the amount of capital or other fixed factors or in the level of technology will lead to an upward shift in the production function (click on the third button) such that a greater level of output may be produced with the same amount of labor input.
Production Possibilities & Opportunity Costs
If we extend our model of production to two (or several) goods, we can develop a more realistic notion of production relationships. In a world of scarce resources, business firms producing different goods are competing for the same pool of factor inputs. In the short run labor is available for the production of one or a combination of goods. However, the desire to increase production of one good 'X' will come at the expense of another good 'Y' as labor or other resources are relallocated from the first good to the second.

In the table below, we can model this competition for resources between two goods: Apples and Bread. In this example, an additional unit of labor directed to bread production allows for producing 25 additional units of bread (the marginal productivity of labor in bread production is constant). Separately, additional units of labor applied to apple production allows for the producing anywhere from 0 to 185 units of apples.

Table 3.

(7 units of Labor available)
 Pt. X(Bread) MPL,Bread Pt. Y(Apples) MPL,Apples A 0 - H 0 - B 25 25 G 37 37 C 50 25 F 74 37 D 75 25 E 110 36 E 100 25 D 140 30 F 125 25 C 163 23 G 150 25 B 179 16 H 175 25 A 185 6

The diagram below summarizes the numbers in the above table. Points on the blue curve -- the Production Possibilities Frontier represent an efficient use of resources. Points within the curve represent inefficient production levels -- resources and technology allow for producing more of good X, good Y, or more of both. Movement along the curve (use the scrollbar to see changes) imply that a tradeoff exists in production when resources are scarce or fixed in supply. Finally points (combinations of the two goods) beyond the frontier are unattainable with existing levels of technology and resource availability.

Figure 2

The Marginal Rate of Transformation
The diagram below defines the slope of this same PPF at any given point. This slope, is known as the Marginal Rate of Transformation (MRT), is a measure of the ratio of marginal productivity's.

Figure 3

Specifically:
MRT = MPy /MPx = Marginal Costx /Marginal Costy

given that: Marginal Costi = wage rate /MPi
This ratio measures the opportunity cost of using resources in producing one good in terms of the alternative use of those resources used in the production of other goods. Given the role of diminishing marginal productivity; as resources are allocated away from good Y towards good X, the opportunity cost (|MRT|), of producing more of good X, increases.

If resources were to be allocated in the opposite direction, the same would be true -- the opportunity cost of producing more of good Y would also increase in terms of foregone production of good X.

Opportunity Costs and Relative Prices
Suppose that we are producing at point D in the above diagrams. If we transfer one unit of labor away from apple production to bread production, we must give up 30 units of apples and gain 25 units of bread. Thus the opportunity "cost" of each unit of bread is 6/5 (1.20) of an apple. Bringing relative prices into the picture, we might find that the price of apples 'Papples' is \$2.00 and the price of bread 'Pbread is \$3.00. Or,
PR = [Px / PY] = [Pbread / Papples] = 1.50.
Stated differently, (as a Terms of Trade), we find that we are willing to trade 1 unit of bread for \$3.00 and with that \$3.00, we could then acquire 1.5 units of apples. One unit of bread is "worth" 1.5 units of apples given these prices.

If we compare the relative price of bread to the opportunity cost of bread, we find that the value of bread in terms of apples is greater than the opportunity cost of producing bread:

one loaf of bread is worth 1.50 apples
one loaf of bread costs 1.20 apples
, or

From a social point of view, we should be allocating more resources towards bread production and away from apple production. With the reallocation of resources, the opportunity cost of bread production will rise. This reallocation should continue until the following is true:
PR = MRT
or

If the price of bread were to fall, say to \$2.00 per unit (click on the Next button), then the relative price of bread would be equal to 1.0 (i.e., each unit of bread is exactly equal to the value of one unit of apples) and PR . With this change, the value of bread, in terms of apples, is less than the opportunity costs of producing bread. In this case we should allocate resources away from bread production (click on the Next button again). This reallocation will cause the opportunity cost of bread to fall until it is just equal to the prevailing relative price. (Click on the Next button again to repeat.)

This example above, demonstrates how the relationship between relative prices (acting as signals about the "value" place on products by consumers) and opportunity costs (that represent the underlying characteristics of production technology) determine an efficient use of resources. As relative prices change, resources will be reallocated to that good that is valued more highly taking into account the opportunity costs of additional production of that good.

Concepts for Review:
• Diminishing Marginal Productivity
• Inefficient Production
• Long Run Production
• Marginal Productivity of Labor
• Marginal Rate of Transformation
• Opportunity Cost
• Production Function
• Production Possibilities
• Relative Prices
• Short Run Production
• Technology
• Unattainable Output Combinations